$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary Let $q: X \rightarrow S$ be a morphism of simplicial sets, where $S$ is a Kan complex. The following conditions are equivalent:


The morphism $q$ is an isofibration.


The morphism $q$ is a cartesian fibration.


The morphism $q$ is a cocartesian fibration.

Proof. We will prove the equivalence $(1) \Leftrightarrow (2)$; the equivalence $(1) \Leftrightarrow (3)$ follows by a similar argument. The implication $(2) \Rightarrow (1)$ is a special case of Proposition For the converse, suppose that $q$ is an isofibration. Then $q$ is an inner fibration. To complete the proof, we must show that for every vertex $y \in X$ and every edge $\overline{e}: \overline{x} \rightarrow q(y)$ of $S$, we can write $\overline{e} = q(e)$ for some $q$-cartesian edge $e$ of $X$. Since $S$ is a Kan complex, $\overline{e}$ is an isomorphism (Proposition Our assumption that $q$ is an isofibration then guarantees that we can write $\overline{e} = q(e)$ for some isomorphism $e: x \rightarrow y$ of $X$. The edge $e$ is automatically $q$-cartesian by virtue of Corollary $\square$